Analysts on harmonic maps over geometric singular spaces via Dirichlet forms

通过狄利克雷形式分析几何奇异空间上的调和映射

基本信息

批准号：
16540201
负责人：
KUWAE Kazuhiro
金额：
$ 2.37万
依托单位：
Kumamoto University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540201/
关键词：
Dirichlet form harmonic map subharmonic function variational convergence Gromov-Hausdorff convergence Alexandrov space Kato class heat kennel アレキサンドロフ空間

项目摘要

We establish the following result :1) Variational convergence of metric measure spaces:We introduce a natural definition of Lp-convergence of maps. $pge 1$, in the case where the domain is a convergent sequence of measured metric space with respect to the measured Gromov-Hausdorff topology and the target is a Gromov-Hausdorff convergent sequence. With the Lp-convergence, we establish a theory of variational convergences. We prove that the Poincare inequality with some additional condition implies the asymptotic compactness. The asymptotic compactness is equivalent to the Gromov-Hausdorff compactness of the energy-sublevel sets. Supposing that the targets are CAT(0)-spaces, we study convergence of resolvents. As applications, we investigate the approximating energy functional over a measured metric space and convergence of energy functionals with a lower bound of Ricci curvature. This work was done with Prof. T. Shioya.2) Perturbation of symmetric Markov processes and its related stocha … More stic calculus:We present a path-space integral representation of the semigroup associated with the quadratic form obtained by a lower orderperturbation of the L2-infinitesimal generator L of a general symmetric Markov process. Using time-reversal, we introduce a stochastic integral for zero-energy additive functionals of symmetric Markov processes, extending earlier work of S. Nakao. Various properties of such stochastic integrals are discussed and an It^o formula for Dirichlet processes is obtained. This work was done with Professors Z.Q. Chen. P.J. Fitzsimmons and T.S. Zhang.3) Kato class measures over symmetric Markov processes :We show that $fin L^p(X ; m)$ implies $|f|dmin S_K^1$ for $p>D$ with $D>0$, where $S_K^1$ is a subfamily of Kato class measures relative to a semigroup kennel $p_t(x, y)$ of a Markov process associated with a (non-symmetric) Dirichlet form on $L^2(X ; m)$. We only assume that $p_t(x, y)$ satisfies the Nash type estimate of small time defending on $D$. No concrete expression of $p_t(X, V)$ is needed for the result. This wonk was done with M. Takahashi.4) Refinements of exceptional sets with respect to (n, p)-capacity oven symmetric Markov processes:We establish a one to one correspondence between a class of smooth measures in the (n, p)-sense and a class of positive continuous additive functionals admitting (n, p)-exceptional sets. This work was done with A. Sato.5) Liouville theorems for harmonic maps to convex spaces over Markov chains:We give a Liouville type theorem for harmonic maps from the space equipped with the harmonicity of functions in terms of conservative Markov chains to convex spaces admitting barycenters. No differentiable structures for the domain and the target are assumed. This work was done with prof. k.Th. Sturm.6) Laplacian comparison theorem on Alexandrov spaces :We consider a directionally restricted version of the Bishop-Gromov relative volume comparison as generalized notion of Ricci curvature bounded below for Alexandrov spaces. We prove a Laplacian comparison theorem for Alexandrov spaces under the condition. As an application we prove a topological splitting theorem. This work was done with Prof. T.Shioya. Less

我们建立以下结果：1）度量测量空间的变异收敛：我们引入了地图的LP连接的自然定义。 $ pge 1 $，如果域是相对于测得的Gromov-Hausdorff拓扑结构的测量公制空间的收敛序列，而目标是Gromov-Hausdorff收敛序列。通过LP - 连接，我们建立了一种变异融合理论。我们证明，具有一些附加条件的繁殖性不平等意味着不对称的紧凑性。不对称的紧凑性等效于能量固定套件的Gromov-Hausdorff紧凑性。假设目标是CAT（0）空间，我们研究已解决的收敛性。作为应用，我们研究了在测得的度量空间上的近似能量功能，并与RICCI曲率的下限的能量函数收敛。这项工作是用T. Shioya.2）进行的，对称对称马尔可夫过程及其相关的stocha…更严格的微积分：我们提出了与较低的二次订购形式相关的半群的路径空间积分表示，这是通过较低的l2 infinitesImal symetric Symmetric Markov进程的较低订购。使用时间逆转，我们引入了一个随机积分，用于对称Markov过程的零能量添加功能，从而扩展了Nakao S.s。nakao的早期工作。讨论了此类随机积分的各种特性，并获得了Dirichlet过程的IT^O公式。这项工作是由Z.Q.教授完成的。陈。 P.J. Fitzsimmons和T.S. Zhang.3) Kato class measures over symmetric Markov processes :We show that $fin L^p(X ; m)$ implies $|f|dmin S_K^1$ for $p>D$ with $D>0$, where $S_K^1$ is a subfamily of Kato class measures relative to a semigroup kennel $p_t(x, y)$ of a Markov process associated with a (non-symmetric) Dirichlet $ l^2（x; m）$上的形式。我们仅假设$ p_t（x，y）$满足$ d $上防御的nash类型估计。结果不需要$ p_t（x，v）$的具体表达。此wonk是用M. takahashi.4）相对于（N，P） - 容量烤箱对称的Markov流程进行的特殊集的改进：我们在（N，P） - sense和一类积极的连续添加功能（N，P）-Exceptional-seption-sense和一类平滑度量之间建立了一对一的对应关系。这项工作是用A. sato.5）liouville定理进行的，用于谐波图，以在马尔可夫链上凸出空间：我们从配备有函数的谐波谐波的谐波图的liouville type定理，以保守的马尔可夫链来谐波链接，以凸出barrycenters。没有假定域和目标的可区分结构。这项工作是由教授完成的。 K.Th. Sturm.6）Alexandrov空间上的Laplacian比较定理：我们将主教Gromov相对体积比较的定向限制版本视为Alexandrov Space的RICCI曲率的概括性通知。我们证明了该条件下Alexandrov空间的Laplacian比较定理。作为应用程序，我们证明了拓扑分解定理。这项工作是由T.Shioya教授完成的。较少的